Law of excluded middle for a statement “Formula A is provable”
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Often in many metamathematical proofs, I see the statement "Either the given formula $A$ is provable in this formal system or it is not". Is it valid for intuitionists as well? For me, it seems a little weird because we are in metamathematics but we still use the law of excluded middle for a potentially infinite number of objects. What is the current stance for this of modern logicians?
logic
asked Jan 13 at 2:47
Daniels KrimansDaniels Krimans
51739
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add a comment |
$begingroup$
Often in many metamathematical proofs, I see the statement "Either the given formula $A$ is provable in this formal system or it is not". Is it valid for intuitionists as well? For me, it seems a little weird because we are in metamathematics but we still use the law of excluded middle for a potentially infinite number of objects. What is the current stance for this of modern logicians?
logic
asked Jan 13 at 2:47
Daniels KrimansDaniels Krimans
51739
$endgroup$
add a comment |
$begingroup$
Often in many metamathematical proofs, I see the statement "Either the given formula $A$ is provable in this formal system or it is not". Is it valid for intuitionists as well? For me, it seems a little weird because we are in metamathematics but we still use the law of excluded middle for a potentially infinite number of objects. What is the current stance for this of modern logicians?
logic
asked Jan 13 at 2:47
Daniels KrimansDaniels Krimans
51739
$endgroup$
Often in many metamathematical proofs, I see the statement "Either the given formula $A$ is provable in this formal system or it is not". Is it valid for intuitionists as well? For me, it seems a little weird because we are in metamathematics but we still use the law of excluded middle for a potentially infinite number of objects. What is the current stance for this of modern logicians?
logic
logic
asked Jan 13 at 2:47
Daniels KrimansDaniels Krimans
51739
asked Jan 13 at 2:47
Daniels KrimansDaniels Krimans
51739
asked Jan 13 at 2:47
Daniels KrimansDaniels Krimans
51739
asked Jan 13 at 2:47
Daniels KrimansDaniels Krimans
51739
asked Jan 13 at 2:47
Daniels KrimansDaniels Krimans
51739
51739
add a comment |
add a comment |
1 Answer
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Unless provability in the formal system $T$ is decidable, intuitionists would not accept the general assertion "For all formulas $A$, either $A$ is provable in $T$ or it is not." For one specific $A$, the assertion "Either $A$ is provable in $T$ or it is not" might be intuitionistically acceptable; it depends on the details of $A$ and $T$.
answered Jan 13 at 4:59
Andreas BlassAndreas Blass
49.6k451108
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thank you for your answer. I am going through one of the Godel's theorem proofs and there the following intuitive predicate is used: $A(a,b):$ $a$ is the Godel's number of formula and $b$ is Godel's number of a proof of the formula. Would that be intuitionistically accepted if $A(a,b)$ is primitive recursive, which means Turing computable?
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– Daniels Krimans
Jan 13 at 6:39
1
$begingroup$
As far as I can see, "provable in $T$ or unprovable in $T$" is intuitionistically correct for primitive recursive predicates, as long as the formal system $T$ includes enough arithmetic; Peano Arithmetic would be more than enough.
$endgroup$
– Andreas Blass
Jan 13 at 15:45
add a comment |
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1 Answer
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1 Answer
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$begingroup$
Unless provability in the formal system $T$ is decidable, intuitionists would not accept the general assertion "For all formulas $A$, either $A$ is provable in $T$ or it is not." For one specific $A$, the assertion "Either $A$ is provable in $T$ or it is not" might be intuitionistically acceptable; it depends on the details of $A$ and $T$.
answered Jan 13 at 4:59
Andreas BlassAndreas Blass
49.6k451108
$endgroup$
$begingroup$
thank you for your answer. I am going through one of the Godel's theorem proofs and there the following intuitive predicate is used: $A(a,b):$ $a$ is the Godel's number of formula and $b$ is Godel's number of a proof of the formula. Would that be intuitionistically accepted if $A(a,b)$ is primitive recursive, which means Turing computable?
$endgroup$
– Daniels Krimans
Jan 13 at 6:39
1
$begingroup$
As far as I can see, "provable in $T$ or unprovable in $T$" is intuitionistically correct for primitive recursive predicates, as long as the formal system $T$ includes enough arithmetic; Peano Arithmetic would be more than enough.
$endgroup$
– Andreas Blass
Jan 13 at 15:45
add a comment |
$begingroup$
Unless provability in the formal system $T$ is decidable, intuitionists would not accept the general assertion "For all formulas $A$, either $A$ is provable in $T$ or it is not." For one specific $A$, the assertion "Either $A$ is provable in $T$ or it is not" might be intuitionistically acceptable; it depends on the details of $A$ and $T$.
answered Jan 13 at 4:59
Andreas BlassAndreas Blass
49.6k451108
$endgroup$
$begingroup$
thank you for your answer. I am going through one of the Godel's theorem proofs and there the following intuitive predicate is used: $A(a,b):$ $a$ is the Godel's number of formula and $b$ is Godel's number of a proof of the formula. Would that be intuitionistically accepted if $A(a,b)$ is primitive recursive, which means Turing computable?
$endgroup$
– Daniels Krimans
Jan 13 at 6:39
1
$begingroup$
As far as I can see, "provable in $T$ or unprovable in $T$" is intuitionistically correct for primitive recursive predicates, as long as the formal system $T$ includes enough arithmetic; Peano Arithmetic would be more than enough.
$endgroup$
– Andreas Blass
Jan 13 at 15:45
add a comment |
$begingroup$
Unless provability in the formal system $T$ is decidable, intuitionists would not accept the general assertion "For all formulas $A$, either $A$ is provable in $T$ or it is not." For one specific $A$, the assertion "Either $A$ is provable in $T$ or it is not" might be intuitionistically acceptable; it depends on the details of $A$ and $T$.
answered Jan 13 at 4:59
Andreas BlassAndreas Blass
49.6k451108
$endgroup$
Unless provability in the formal system $T$ is decidable, intuitionists would not accept the general assertion "For all formulas $A$, either $A$ is provable in $T$ or it is not." For one specific $A$, the assertion "Either $A$ is provable in $T$ or it is not" might be intuitionistically acceptable; it depends on the details of $A$ and $T$.
answered Jan 13 at 4:59
Andreas BlassAndreas Blass
49.6k451108
answered Jan 13 at 4:59
Andreas BlassAndreas Blass
49.6k451108
answered Jan 13 at 4:59
Andreas BlassAndreas Blass
49.6k451108
answered Jan 13 at 4:59
Andreas BlassAndreas Blass
49.6k451108
49.6k451108
$begingroup$
thank you for your answer. I am going through one of the Godel's theorem proofs and there the following intuitive predicate is used: $A(a,b):$ $a$ is the Godel's number of formula and $b$ is Godel's number of a proof of the formula. Would that be intuitionistically accepted if $A(a,b)$ is primitive recursive, which means Turing computable?
$endgroup$
– Daniels Krimans
Jan 13 at 6:39
1
$begingroup$
As far as I can see, "provable in $T$ or unprovable in $T$" is intuitionistically correct for primitive recursive predicates, as long as the formal system $T$ includes enough arithmetic; Peano Arithmetic would be more than enough.
$endgroup$
– Andreas Blass
Jan 13 at 15:45
add a comment |
$begingroup$
thank you for your answer. I am going through one of the Godel's theorem proofs and there the following intuitive predicate is used: $A(a,b):$ $a$ is the Godel's number of formula and $b$ is Godel's number of a proof of the formula. Would that be intuitionistically accepted if $A(a,b)$ is primitive recursive, which means Turing computable?
$endgroup$
– Daniels Krimans
Jan 13 at 6:39
1
$begingroup$
As far as I can see, "provable in $T$ or unprovable in $T$" is intuitionistically correct for primitive recursive predicates, as long as the formal system $T$ includes enough arithmetic; Peano Arithmetic would be more than enough.
$endgroup$
– Andreas Blass
Jan 13 at 15:45
$begingroup$
thank you for your answer. I am going through one of the Godel's theorem proofs and there the following intuitive predicate is used: $A(a,b):$ $a$ is the Godel's number of formula and $b$ is Godel's number of a proof of the formula. Would that be intuitionistically accepted if $A(a,b)$ is primitive recursive, which means Turing computable?
$endgroup$
– Daniels Krimans
Jan 13 at 6:39
$begingroup$
thank you for your answer. I am going through one of the Godel's theorem proofs and there the following intuitive predicate is used: $A(a,b):$ $a$ is the Godel's number of formula and $b$ is Godel's number of a proof of the formula. Would that be intuitionistically accepted if $A(a,b)$ is primitive recursive, which means Turing computable?
$endgroup$
– Daniels Krimans
Jan 13 at 6:39
1
1
$begingroup$
As far as I can see, "provable in $T$ or unprovable in $T$" is intuitionistically correct for primitive recursive predicates, as long as the formal system $T$ includes enough arithmetic; Peano Arithmetic would be more than enough.
$endgroup$
– Andreas Blass
Jan 13 at 15:45
$begingroup$
As far as I can see, "provable in $T$ or unprovable in $T$" is intuitionistically correct for primitive recursive predicates, as long as the formal system $T$ includes enough arithmetic; Peano Arithmetic would be more than enough.
$endgroup$
– Andreas Blass
Jan 13 at 15:45
add a comment |
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